Sequential Processing of Integrated Measurements in a Generalized Kalman Filter

David Hartman

Abstract:	In reference 1, the author presented a new optimal algorithm for processing GPS delta pseudo range measurements. Many Kalman Filter algorithms in common use are suboptimal in that the process noise terms are treated in an approximate manner. The Kalman Filter algorithm relates measurements to states at the current time only. Delta pseudo range measurements are integrated from carrier phase information over an interval and strictly are related to states at the beginning and end of the integration interval. The algorithm presented in reference 1 showed that it is possible to process these integrated measurements optimally without carrying additional states for quantities at the beginning of the integration interval. The Kalman Gain and covariance update equations of the Kalman Filter were generalized to properly account for the integrated process noise over the integration interval. That algorithm was valid for vector updates of delta pseudo range measurements. Kalman Filter algorithms often process a vector of measurements as a sequence of scalar updates to avoid a matrix inversion in the computation of the Kalman gain. This paper addresses the scalar update case for the algorithm of reference 1. In general, it is not possible to optimally process integrated measurements sequentially without estimating additional states which relate to the integrated process noise over the integration interval. The general algorithm for sequential processing of integrated measurements is presented in this paper. In an important special case, integrated measurements can be processed sequentially without estimating any additional states. For this special case, the measurement noise covariance matrix R must be diagonal, another covariance matrix designated Pq related to integrated process noise must be diagonal, and certain other restrictions must be obeyed. These restrictions insure that the optimal prediction of the integrated process noise terms’ contribution to each sequential measurement residual is zero. This means that no explicit estimation of the process noise terms need be implemented. The matrix Pq is rarely diagonal. For the Kalman Filter, the state estimates that are optimal for a set of measurements are also optimal for any invertible linear combination of those measurements. Because of this property, a similarity transformation can be applied to find a new set of measurements for which the transformed Pq’ is diagonal. The transformation which diagonalizes Pq will not leave the transformed measurement noise covariance R’ diagonal except in the case where R has the form R=ó2 I, where ó2 is a scalar constant and I is the identity matrix. To obey the additional constraints, two measurement matrices for the integrated measurements must couple to disjoint parts of the state space. If R has this special form, and the two measurement matrices have the special form, then integrated measurements can be readily processed sequentially without estimating other states.
Published in:	Proceedings of the 2002 National Technical Meeting of The Institute of Navigation January 28 - 30, 2002 The Catamaran Resort Hotel San Diego, CA
Pages:	632 - 638
Cite this article:	Hartman, David, "Sequential Processing of Integrated Measurements in a Generalized Kalman Filter," Proceedings of the 2002 National Technical Meeting of The Institute of Navigation, San Diego, CA, January 2002, pp. 632-638.
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